Post-Newtonian Mechanics of the Earth-Moon System
2Department of Physics and Astronomy
University of Missouri-Columbia, USA
Journées 2010, 20-22 September 2010
Yi Xie1,2, Sergei Kopeikin2
1Astronomy Department
Nanjing University, China
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Outline
Motivations
Post-Newtonian reference frames
Post-Newtonian mechanics
Outlook
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Motivations
Distinguish physics from coordinates
Comprehensive and self-consistent theory of PN Equations of Motion
Current and future high-precision experiments
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MotivationsDistinguish physics from coordinates
Curved spacetime:
Coordinates
Physics
Strongly coupledDifficult to disentangle
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ai=aiNewton
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c2∑j≠i
vi× vj×GMj
rij3 rji⋯
Extrinsic gravitomagnetic effects:
PPN-EIH:
t '=t− 1
c 4∑B
B
GMB
rB
rB⋅vB
x '=x−1c 2∑
B
B
GMB
rB
rB
∑j≠i
22−j
c2 vi× vj×GMj
rij3 rji
gravitomagnetic force:
depends on the choice of coordinates .j
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MotivationsComprehensive theory of PN EoMEinstein-Infeld-Hoffman: N-point particleBarker & O'Connell: spin (point particle)Brumberg-Kopeikin: N-body (Newtonian multipoles+spins)Damour-Soffel-Xu: N-body (PN multipoles+spins in GR)Kopeikin & Vlasov: N-body (PN multipole+spins in STT)Racine & Flanagan: extend BK+DSX (flux integral method)Xie & Kopeikin: Earth-Moon reference frames (offprints available)
Our goals:
PN EoM with multipoles expressed explicitly with coordinates and velocities of bodies
MBaB=F (xC, vC; IC<L>, SC
<L>;γ,β)
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Motivations
Current and future high-precision experiments:Lunar Laser Ranging
relativisitic parameters selenophysics: liquid core?
Pulsar timingGravitational wave astronomy:
Earth-Moon system as a detector of GWs
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EMBSSB
GRF
SRF
Ĝμν(Σ,Y)
gμν(t,x)
Gμν(T,X )
ĝμν(u,w)
PN Reference Frames
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PN Reference FramesA global frame
Solar System Barycentric Frame orbital motions
N local frames
eg. Geocentric Frame internal motions of matter, time evolution of multipoles
Some local frames for subsystems
eg. the Earth-Moon Barycentric Frame relative motions for LLR
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e=0.00549
Why local frames? Multipole moments
S
Ev
M
Lorentzcontraction~2 m (Lunar Orbit)
S
E
M
Einsteincontraction
~4 m (Lunar Orbit)
Earth+Moon as acomposite body
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e=0.00549
Why to use the local frame for subsystems?
Hierarchical systems: extension of IAU2000 resolutions
Universe -> superclusters -> clusters -> groups
-> galaxies -> stellar clusters -> stellar systems
-> planetary systems
Lunar mass is not negligible: M/E~1/80
The Geocentric Frame can not cover the Moon.
LLR is a local frame experiment
EMB frame helps to understand the gauge freedom
of equations of motion.
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Momentum centered
I=0
A coincides with B (a constraint
on dipole)
dI/du=0I=Pu+I0
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PN Mechanics
PN orbital motion (external problem)
the body B's motion in the SSB frame
PN relative motion (internal problem)
Moon with respect to Earth in the EMB frame
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PN orbital motion Conservation of dipole moment: d2I/du2=0
Matching procedure: coordinate transformation, the law of motion of the origin of the local frame
... ...
The PPN translational equations of motion of extended bodies in the global frame
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2-body interactions: RBC 3-body interactions: RBCRBD, RBCRCD
spin effects
B
CD
PN orbital motion
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Definition of Mass (Blanchet-Damour)
1. a single particle: rest mass
2. a single body: rest mass + internal energy
3. a body in N-body system: rest mass + internal energy + external effects
rest mass internal energy external effects
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MB≠IB inertial mass ≠ gravitational mass violation of Strong EquivalencePrinciple in Scalar-Tensor theory
Violation of Strong Equivalence Principle
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PN relative Motion
PN orbital motion of Moon in EMB
PN orbital motion of Earth in EMB
Moon-Earth in the EMB frame
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Outlook: SelenophysicsFluid coremore convincing evidences, radius
Elastic propertiesLove numbers
Spin-orbit coupling
Harmonics in the multi-layer Moon
Dissipation and convection
Free libration modes
Topography and mascons
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Outlook: Fundamental PhysicsStronger limitationsnonlinear parameter , dark energy, scalar fieldsβ
New relativistic effectsrelativistic precession of lunar orbit w.r.t. ICRFrelativistic quadrupole moment of the Earthtidal gravitomagnetic effectsviolation of SEP
Stochastic GW background
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Thank you!
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